On the failure of the chain rule for the divergence of Sobolev vector fields

TL;DR

This paper demonstrates the failure of the chain rule for divergence in Sobolev vector fields by constructing specific examples where the expected divergence properties do not hold, especially in higher dimensions.

Contribution

It constructs a broad class of Sobolev incompressible vector fields showing the chain rule for divergence can fail under certain renormalization conditions.

Findings

Counterexamples to the chain rule for divergence in Sobolev spaces

Failure occurs for a wide class of renormalization maps and defects

Results hold in dimensions d ≥ 3

Abstract

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = {\rm div}\, h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $\rho \in L^p$ for which ${\rm div}\, (\rho u) =0$ but ${\rm div}\, (\beta(\rho) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$